Integrand size = 27, antiderivative size = 116 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=2 a^2 x+\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
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Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2951, 3855, 3852, 8, 3853, 2718} \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}+\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{8 d}+2 a^2 x \]
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Rule 8
Rule 2718
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (2 a^6-a^6 \csc (c+d x)-4 a^6 \csc ^2(c+d x)-a^6 \csc ^3(c+d x)+2 a^6 \csc ^4(c+d x)+a^6 \csc ^5(c+d x)+a^6 \sin (c+d x)\right ) \, dx}{a^4} \\ & = 2 a^2 x-a^2 \int \csc (c+d x) \, dx-a^2 \int \csc ^3(c+d x) \, dx+a^2 \int \csc ^5(c+d x) \, dx+a^2 \int \sin (c+d x) \, dx+\left (2 a^2\right ) \int \csc ^4(c+d x) \, dx-\left (4 a^2\right ) \int \csc ^2(c+d x) \, dx \\ & = 2 a^2 x+\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{2} a^2 \int \csc (c+d x) \, dx+\frac {1}{4} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (4 a^2\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = 2 a^2 x+\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = 2 a^2 x+\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}
Time = 6.65 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.85 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (192 \cot (c+d x)+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (8+3 \csc (c+d x))-2 \csc ^2\left (\frac {1}{2} (c+d x)\right ) (64+3 \csc (c+d x))-24 \csc (c+d x) \left (16 (c+d x)+9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+8 (7+8 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )+24 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-48 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x) (1+\sin (c+d x))^2}{192 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
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Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(168\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(168\) |
parallelrisch | \(\frac {a^{2} \left (-216 \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d -3 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-224 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 d x -3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+224 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(184\) |
risch | \(2 a^{2} x -\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {a^{2} \left (3 \,{\mathrm e}^{7 i \left (d x +c \right )}+21 \,{\mathrm e}^{5 i \left (d x +c \right )}-96 i {\mathrm e}^{6 i \left (d x +c \right )}+21 \,{\mathrm e}^{3 i \left (d x +c \right )}+192 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-160 i {\mathrm e}^{2 i \left (d x +c \right )}+64 i\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(186\) |
norman | \(\frac {-\frac {a^{2}}{64 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d}-\frac {a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {13 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {7 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {7 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {13 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+2 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {65 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {65 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {9 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(314\) |
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (108) = 216\).
Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.89 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {96 \, a^{2} d x \cos \left (d x + c\right )^{4} - 48 \, a^{2} \cos \left (d x + c\right )^{5} - 192 \, a^{2} d x \cos \left (d x + c\right )^{2} + 90 \, a^{2} \cos \left (d x + c\right )^{3} + 96 \, a^{2} d x - 54 \, a^{2} \cos \left (d x + c\right ) + 27 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 27 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.44 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {32 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} - 3 \, a^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.40 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, {\left (d x + c\right )} a^{2} - 216 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {384 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 10.01 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.28 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {9\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {4\,a^2\,\mathrm {atan}\left (\frac {16\,a^4}{9\,a^4+16\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {9\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9\,a^4+16\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,d}-\frac {-20\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {56\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a^2}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )} \]
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